Retractions of split graphs and End-orthodox split graphs

A retraction f of a graph G is an edge-preserving mapping of G with f(v)=v for all v∈V(H), where H is the subgraph induced by the range of f. A graph G is called End-orthodox (End-regular) if its endomorphism monoid End X is orthodox (regular) in the semigroup sense. It is known that a graph is End-orthodox if it is End-regular and the composition of any two retractions is also a retraction. The retractions of split graphs are given and End-orthodox split graphs are characterized.